Construct Single-Hierarchical P/NBD Model for Short Timeframe Synthetic Data
In this workbook we construct the non-hierarchical P/NBD models on the synthetic data with the longer timeframe.
1 Load and Construct Datasets
We start by modelling the P/NBD model using our synthetic datasets before we try to model real-life data.
Show code
use_fit_start_date <- as.Date("2020-01-01")
use_fit_end_date <- as.Date("2022-01-01")
use_valid_start_date <- as.Date("2022-01-01")
use_valid_end_date <- as.Date("2023-01-01")1.1 Load Long Time-frame Synthetic Data
We now want to load the short time-frame synthetic data.
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customer_cohortdata_tbl <- read_rds("data/synthdata_shortframe_cohort_tbl.rds")
customer_cohortdata_tbl |> glimpse()Rows: 50,000
Columns: 4
$ customer_id <chr> "SFC202001_0001", "SFC202001_0002", "SFC202001_0003", "…
$ cohort_qtr <chr> "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1", …
$ cohort_ym <chr> "2020 01", "2020 01", "2020 01", "2020 01", "2020 01", …
$ first_tnx_date <date> 2020-01-01, 2020-01-01, 2020-01-01, 2020-01-01, 2020-0…
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customer_simparams_tbl <- read_rds("data/synthdata_shortframe_simparams_tbl.rds")
customer_simparams_tbl |> glimpse()Rows: 50,000
Columns: 9
$ customer_id <chr> "SFC202001_0001", "SFC202001_0002", "SFC202001_0003", …
$ cohort_qtr <chr> "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1",…
$ cohort_ym <chr> "2020 01", "2020 01", "2020 01", "2020 01", "2020 01",…
$ first_tnx_date <date> 2020-01-01, 2020-01-01, 2020-01-01, 2020-01-01, 2020-…
$ customer_lambda <dbl> 0.12213805, 0.29987747, 0.31504009, 0.03856001, 0.1881…
$ customer_mu <dbl> 0.127118566, 0.096184402, 0.052334526, 0.204708842, 0.…
$ customer_tau <dbl> 29.5956480, 10.9437199, 1.6938450, 2.6798108, 48.75206…
$ customer_amtmn <dbl> 46.2371662, 111.0425353, 45.8870891, 43.0754249, 10.93…
$ customer_amtcv <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
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customer_transactions_tbl <- read_rds("data/synthdata_shortframe_transactions_tbl.rds")
customer_transactions_tbl |> glimpse()Rows: 297,815
Columns: 4
$ customer_id <chr> "SFC202001_0028", "SFC202001_0006", "SFC202001_0012", "S…
$ tnx_timestamp <dttm> 2020-01-01 00:59:15, 2020-01-01 01:25:43, 2020-01-01 04…
$ invoice_id <chr> "T20200101-0001", "T20200101-0002", "T20200101-0003", "T…
$ tnx_amount <dbl> 269.07, 124.62, 20.81, 14.76, 16.02, 19.16, 211.91, 43.9…
1.2 Load Derived Data
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id_1000 <- read_rds("data/shortsynth_id_1000.rds")
id_5000 <- read_rds("data/shortsynth_id_5000.rds")
id_10000 <- read_rds("data/shortsynth_id_10000.rds")
fit_1000_data_tbl <- read_rds("data/shortsynth_fit_1000_data_tbl.rds")
fit_10000_data_tbl <- read_rds("data/shortsynth_fit_10000_data_tbl.rds")
customer_fit_stats_tbl <- fit_1000_data_tbl
customer_summarystats_tbl <- read_rds("data/shortsynth_customer_summarystats_tbl.rds")
obs_fitdata_tbl <- read_rds("data/shortsynth_obs_fitdata_tbl.rds")
obs_validdata_tbl <- read_rds("data/shortsynth_obs_validdata_tbl.rds")Finally, we need to set our directories where we save our Stan code and the model outputs.
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stan_modeldir <- "stan_models"
stan_codedir <- "stan_code"2 Fit First Hierarchical Lambda Model
Our first hierarchical model puts a hierarchical prior around the mean of our population \(\lambda\) - lambda_mn.
Once again we use a Gamma prior for it.
2.1 Compile and Fit Stan Model
We now compile this model using CmdStanR.
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pnbd_onehierlambda_stanmodel <- cmdstan_model(
"stan_code/pnbd_onehier_lambda.stan",
include_paths = stan_codedir,
pedantic = TRUE,
dir = stan_modeldir
)We then use this compiled model with our data to produce a fit of the data.
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stan_modelname <- "pnbd_init_short_onehierlambda1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_lambda_mn_p1 = 0.25,
hier_lambda_mn_p2 = 1,
lambda_cv = 1.00,
mu_mn = 0.10,
mu_cv = 1.00,
)
if(!file_exists(stanfit_object_file)) {
pnbd_init_short_onehierlambda1_stanfit <- pnbd_onehierlambda_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_init_short_onehierlambda1_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_init_short_onehierlambda1_stanfit <- read_rds(stanfit_object_file)
}
pnbd_init_short_onehierlambda1_stanfit$summary()# A tibble: 3,003 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -1.47e+4 -1.47e+4 35.0 34.7 -1.47e+4 -1.46e+4 1.01 472.
2 lambda_mn 2.63e-1 2.62e-1 0.0111 0.0111 2.45e-1 2.82e-1 1.00 2614.
3 lambda[1] 6.66e-1 6.62e-1 0.0761 0.0729 5.49e-1 7.97e-1 1.00 5065.
4 lambda[2] 7.61e-1 7.47e-1 0.202 0.200 4.61e-1 1.12e+0 1.00 5072.
5 lambda[3] 1.44e-1 8.09e-2 0.170 0.0932 4.79e-3 4.91e-1 0.999 2544.
6 lambda[4] 2.89e-1 2.30e-1 0.224 0.180 3.96e-2 7.43e-1 1.00 3157.
7 lambda[5] 5.89e-2 5.26e-2 0.0326 0.0296 1.72e-2 1.23e-1 1.00 3255.
8 lambda[6] 1.46e-1 8.42e-2 0.176 0.0976 4.69e-3 5.03e-1 0.999 2197.
9 lambda[7] 5.06e-1 4.88e-1 0.187 0.174 2.36e-1 8.33e-1 1.00 3462.
10 lambda[8] 2.08e-1 1.89e-1 0.108 0.0997 6.99e-2 4.14e-1 1.00 4389.
# ℹ 2,993 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_init_short_onehierlambda1_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda1-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
2.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"lambda_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_init_short_onehierlambda1_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_init_short_onehierlambda1_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")2.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_init_short_onehierlambda1_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_init_short_onehierlambda1_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_init_short_onehierlambda1",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_init_short_onehierlambda1_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_init_short_onehierlambda1_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_init_short_onehierlambda1_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_init_short_onehierlambda1_assess_valid_simstats_tbl.rds"
2.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_init_short_onehierlambda1_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
2.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_init_short_onehierlambda1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
3 Fit Second Hierarchical Lambda Model
In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.
3.1 Fit Stan Model
We now want to fit the model to our data using this alternative prior for lambda_mn.
Show code
stan_modelname <- "pnbd_init_short_onehierlambda2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_lambda_mn_p1 = 0.50,
hier_lambda_mn_p2 = 1,
lambda_cv = 1.00,
mu_mn = 0.10,
mu_cv = 1.00,
)
if(!file_exists(stanfit_object_file)) {
pnbd_init_short_onehierlambda2_stanfit <- pnbd_onehierlambda_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_init_short_onehierlambda2_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_init_short_onehierlambda2_stanfit <- read_rds(stanfit_object_file)
}
pnbd_init_short_onehierlambda2_stanfit$summary()# A tibble: 3,003 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -1.47e+4 -1.46e+4 34.6 33.8 -1.47e+4 -1.46e+4 1.00 757.
2 lambda_mn 2.63e-1 2.63e-1 0.0113 0.0115 2.45e-1 2.82e-1 1.00 2595.
3 lambda[1] 6.67e-1 6.63e-1 0.0767 0.0744 5.50e-1 8.00e-1 0.999 4817.
4 lambda[2] 7.59e-1 7.44e-1 0.194 0.195 4.72e-1 1.09e+0 1.00 4384.
5 lambda[3] 1.48e-1 8.48e-2 0.177 0.0987 4.05e-3 5.04e-1 1.00 2912.
6 lambda[4] 2.91e-1 2.21e-1 0.246 0.184 3.73e-2 7.73e-1 1.00 4189.
7 lambda[5] 5.83e-2 5.15e-2 0.0321 0.0273 1.80e-2 1.19e-1 0.999 3609.
8 lambda[6] 1.44e-1 8.47e-2 0.170 0.0980 5.04e-3 4.81e-1 1.00 3119.
9 lambda[7] 5.09e-1 4.84e-1 0.189 0.181 2.49e-1 8.57e-1 1.00 3730.
10 lambda[8] 2.10e-1 1.90e-1 0.108 0.101 7.08e-2 4.12e-1 1.00 3847.
# ℹ 2,993 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_init_short_onehierlambda2_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda2-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
3.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"lambda_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_init_short_onehierlambda2_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_init_short_onehierlambda2_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")3.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_init_short_onehierlambda2_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_init_short_onehierlambda2_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_init_short_onehierlambda2",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_init_short_onehierlambda2_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_init_short_onehierlambda2_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_init_short_onehierlambda2_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_init_short_onehierlambda2_assess_valid_simstats_tbl.rds"
3.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_init_short_onehierlambda2_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
3.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_init_short_onehierlambda1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
4 Fit First Hierarchical Mu Model
We now construct the same hierarchical model but based around mu_mn.
4.1 Compile and Fit Stan Model
We compile this model using CmdStanR.
Show code
pnbd_onehiermu_stanmodel <- cmdstan_model(
"stan_code/pnbd_onehier_mu.stan",
include_paths = stan_codedir,
pedantic = TRUE,
dir = stan_modeldir
)We then use this compiled model with our data to produce a fit of the data.
Show code
stan_modelname <- "pnbd_init_short_onehiermu1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_mu_mn_p1 = 0.50,
hier_mu_mn_p2 = 1.00,
lambda_mn = 0.25,
lambda_cv = 1.00,
mu_cv = 1.00
)
if(!file_exists(stanfit_object_file)) {
pnbd_init_short_onehiermu1_stanfit <- pnbd_onehiermu_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_init_short_onehiermu1_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_init_short_onehiermu1_stanfit <- read_rds(stanfit_object_file)
}
pnbd_init_short_onehiermu1_stanfit$summary()# A tibble: 3,003 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -1.37e+4 -1.37e+4 3.59e+1 3.57e+1 -1.38e+4 -1.37e+4 1.01 531.
2 mu_mn 1.06e-1 1.05e-1 7.26e-3 7.20e-3 9.42e-2 1.18e-1 1.00 891.
3 lambda[1] 6.65e-1 6.61e-1 7.55e-2 7.79e-2 5.49e-1 7.96e-1 1.00 4584.
4 lambda[2] 7.51e-1 7.33e-1 2.00e-1 1.91e-1 4.62e-1 1.12e+0 1.00 4930.
5 lambda[3] 1.46e-1 8.25e-2 1.85e-1 9.49e-2 4.90e-3 4.88e-1 1.00 2876.
6 lambda[4] 2.90e-1 2.32e-1 2.35e-1 1.96e-1 3.93e-2 7.56e-1 1.00 3733.
7 lambda[5] 5.81e-2 5.27e-2 3.16e-2 2.87e-2 1.76e-2 1.20e-1 1.00 4621.
8 lambda[6] 1.47e-1 8.49e-2 1.85e-1 9.88e-2 4.68e-3 5.07e-1 0.999 3456.
9 lambda[7] 5.02e-1 4.74e-1 1.86e-1 1.79e-1 2.45e-1 8.58e-1 1.00 5024.
10 lambda[8] 2.07e-1 1.87e-1 1.13e-1 1.02e-1 6.11e-2 4.11e-1 1.00 3959.
# ℹ 2,993 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_init_short_onehiermu1_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu1-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
4.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"mu_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_init_short_onehiermu1_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_init_short_onehiermu1_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")4.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_init_short_onehiermu1_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_init_short_onehiermu1_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_init_short_onehiermu1",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_init_short_onehiermu1_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_init_short_onehiermu1_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_init_short_onehiermu1_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_init_short_onehiermu1_assess_valid_simstats_tbl.rds"
4.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_init_short_onehiermu1_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
4.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_init_short_onehierlambda1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
5 Fit Second Hierarchical Lambda Model
In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.
5.1 Fit Stan Model
We now want to fit the model to our data using this alternative prior for lambda_mn.
Show code
stan_modelname <- "pnbd_init_short_onehiermu2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_mu_mn_p1 = 0.25,
hier_mu_mn_p2 = 1.00,
lambda_mn = 0.25,
lambda_cv = 1.00,
mu_cv = 1.00
)
if(!file_exists(stanfit_object_file)) {
pnbd_init_short_onehiermu2_stanfit <- pnbd_onehiermu_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_init_short_onehiermu2_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_init_short_onehiermu2_stanfit <- read_rds(stanfit_object_file)
}
pnbd_init_short_onehiermu2_stanfit$summary()# A tibble: 3,003 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -1.37e+4 -1.37e+4 3.59e+1 3.46e+1 -1.38e+4 -1.37e+4 1.00 498.
2 mu_mn 1.05e-1 1.05e-1 7.64e-3 7.50e-3 9.33e-2 1.19e-1 1.00 914.
3 lambda[1] 6.67e-1 6.63e-1 7.94e-2 7.69e-2 5.43e-1 8.03e-1 1.01 5316.
4 lambda[2] 7.52e-1 7.38e-1 1.98e-1 2.02e-1 4.53e-1 1.11e+0 1.00 5029.
5 lambda[3] 1.45e-1 8.19e-2 1.77e-1 9.28e-2 4.84e-3 4.97e-1 1.00 2854.
6 lambda[4] 2.83e-1 2.15e-1 2.39e-1 1.81e-1 3.63e-2 7.29e-1 1.00 3342.
7 lambda[5] 5.87e-2 5.34e-2 3.12e-2 2.83e-2 1.88e-2 1.19e-1 1.00 3307.
8 lambda[6] 1.42e-1 8.65e-2 1.73e-1 9.67e-2 4.78e-3 4.68e-1 1.00 3498.
9 lambda[7] 5.04e-1 4.78e-1 1.90e-1 1.66e-1 2.39e-1 8.71e-1 1.00 5589.
10 lambda[8] 2.07e-1 1.85e-1 1.13e-1 1.04e-1 6.61e-2 4.20e-1 1.00 4643.
# ℹ 2,993 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_init_short_onehiermu2_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu2-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
5.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"mu_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_init_short_onehiermu2_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_init_short_onehiermu2_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")5.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_init_short_onehiermu2_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_init_short_onehiermu2_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_init_short_onehiermu2",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_init_short_onehiermu2_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_init_short_onehiermu2_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_init_short_onehiermu2_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_init_short_onehiermu2_assess_valid_simstats_tbl.rds"
5.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_init_short_onehiermu2_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
5.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_init_short_onehiermu1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
6 Compare Model Outputs
We have looked at each of the models individually, but it is also worth looking at each of the models as a group.
Show code
calculate_simulation_statistics <- function(file_rds) {
simdata_tbl <- read_rds(file_rds)
multicount_cust_tbl <- simdata_tbl |>
filter(sim_tnx_count > 0) |>
count(draw_id, name = "multicust_count")
totaltnx_data_tbl <- simdata_tbl |>
count(draw_id, wt = sim_tnx_count, name = "simtnx_count")
simstats_tbl <- multicount_cust_tbl |>
inner_join(totaltnx_data_tbl, by = "draw_id")
return(simstats_tbl)
}Show code
obs_fit_customer_count <- obs_fitdata_tbl |>
filter(tnx_count > 0) |>
nrow()
obs_valid_customer_count <- obs_validdata_tbl |>
filter(tnx_count > 0) |>
nrow()
obs_fit_total_count <- obs_fitdata_tbl |>
pull(tnx_count) |>
sum()
obs_valid_total_count <- obs_validdata_tbl |>
pull(tnx_count) |>
sum()
obs_stats_tbl <- tribble(
~assess_type, ~name, ~obs_value,
"fit", "multicust_count", obs_fit_customer_count,
"fit", "simtnx_count", obs_fit_total_count,
"valid", "multicust_count", obs_valid_customer_count,
"valid", "simtnx_count", obs_valid_total_count
)
model_assess_tbl <- dir_ls("data", regexp = "pnbd_init_short_(one|fixed).*_assess") |>
enframe(name = NULL, value = "file_path") |>
filter(str_detect(file_path, "_assess_model_", negate = TRUE)) |>
mutate(
model_label = str_replace(file_path, "data/pnbd_init_short_(.*?)_assess_.*", "\\1"),
assess_type = if_else(str_detect(file_path, "_assess_fit_"), "fit", "valid"),
sim_data = map(
file_path, calculate_simulation_statistics,
.progress = "calculate_simulation_statistics"
)
)
model_assess_tbl |> glimpse()Rows: 14
Columns: 4
$ file_path <fs::path> "data/pnbd_init_short_fixed1_assess_fit_simstats_tbl.…
$ model_label <chr> "fixed1", "fixed1", "fixed2", "fixed2", "fixed3", "fixed3"…
$ assess_type <chr> "fit", "valid", "fit", "valid", "fit", "valid", "fit", "va…
$ sim_data <list> [<tbl_df[2000 x 3]>], [<tbl_df[2000 x 3]>], [<tbl_df[2000…
Show code
model_assess_summstat_tbl <- model_assess_tbl |>
select(model_label, assess_type, sim_data) |>
unnest(sim_data) |>
pivot_longer(
cols = !c(model_label, assess_type, draw_id)
) |>
group_by(model_label, assess_type, name) |>
summarise(
.groups = "drop",
mean_val = mean(value),
p10 = quantile(value, 0.10),
p25 = quantile(value, 0.25),
p50 = quantile(value, 0.50),
p75 = quantile(value, 0.75),
p90 = quantile(value, 0.90)
)
model_assess_summstat_tbl |> glimpse()Rows: 28
Columns: 9
$ model_label <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed2", "fixed2"…
$ assess_type <chr> "fit", "fit", "valid", "valid", "fit", "fit", "valid", "va…
$ name <chr> "multicust_count", "simtnx_count", "multicust_count", "sim…
$ mean_val <dbl> 620.9155, 4283.5425, 182.8110, 1862.9220, 680.4490, 3177.8…
$ p10 <dbl> 602.0, 3993.9, 172.0, 1674.0, 661.0, 2981.0, 129.9, 828.0,…
$ p25 <dbl> 611.00, 4118.75, 177.00, 1768.00, 671.00, 3067.00, 134.00,…
$ p50 <dbl> 621.0, 4279.5, 183.0, 1857.0, 681.0, 3175.0, 139.0, 939.0,…
$ p75 <dbl> 631.00, 4437.00, 189.00, 1962.25, 690.00, 3285.25, 144.00,…
$ p90 <dbl> 639.0, 4580.0, 194.0, 2051.0, 698.1, 3384.0, 149.0, 1072.0…
Show code
#! echo: TRUE
ggplot(model_assess_summstat_tbl) +
geom_errorbar(
aes(x = model_label, ymin = p10, ymax = p90), width = 0
) +
geom_errorbar(
aes(x = model_label, ymin = p25, ymax = p75), width = 0, linewidth = 3
) +
geom_hline(
aes(yintercept = obs_value),
data = obs_stats_tbl, colour = "red"
) +
scale_y_continuous(labels = label_comma()) +
expand_limits(y = 0) +
facet_wrap(
vars(assess_type, name), scale = "free_y"
) +
labs(
x = "Model",
y = "Count",
title = "Comparison Plot for the Different Models"
) +
theme(
axis.text.x = element_text(angle = 20, vjust = 0.5, size = 8)
)6.1 Write Assessment Data to Disk
We now want to save the assessment data to disk.
Show code
model_assess_tbl |> write_rds("data/assess_data_pnbd_short_onehier_tbl.rds")7 R Environment
Show code
options(width = 120L)
sessioninfo::session_info()─ Session info ───────────────────────────────────────────────────────────────────────────────────────────────────────
setting value
version R version 4.2.3 (2023-03-15)
os Ubuntu 22.04.2 LTS
system x86_64, linux-gnu
ui X11
language (EN)
collate en_US.UTF-8
ctype en_US.UTF-8
tz Europe/Dublin
date 2023-06-09
pandoc 2.19.2 @ /usr/local/bin/ (via rmarkdown)
─ Packages ───────────────────────────────────────────────────────────────────────────────────────────────────────────
package * version date (UTC) lib source
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arrayhelpers 1.1-0 2020-02-04 [1] RSPM (R 4.2.0)
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boot 1.3-28.1 2022-11-22 [2] CRAN (R 4.2.3)
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rstan 2.21.8 2023-01-17 [1] RSPM (R 4.2.0)
rstantools 2.3.1 2023-03-30 [1] RSPM (R 4.2.0)
rsyslog * 1.0.2 2021-06-04 [1] RSPM (R 4.2.0)
scales * 1.2.1 2022-08-20 [1] RSPM (R 4.2.0)
sessioninfo 1.2.2 2021-12-06 [1] RSPM (R 4.2.0)
shiny 1.7.4 2022-12-15 [1] RSPM (R 4.2.0)
shinyjs 2.1.0 2021-12-23 [1] RSPM (R 4.2.0)
shinystan 2.6.0 2022-03-03 [1] RSPM (R 4.2.0)
shinythemes 1.2.0 2021-01-25 [1] RSPM (R 4.2.0)
StanHeaders 2.21.0-7 2020-12-17 [1] RSPM (R 4.2.0)
stringi 1.7.12 2023-01-11 [1] RSPM (R 4.2.0)
stringr * 1.5.0 2022-12-02 [1] RSPM (R 4.2.0)
svUnit 1.0.6 2021-04-19 [1] RSPM (R 4.2.0)
tensorA 0.36.2 2020-11-19 [1] RSPM (R 4.2.0)
threejs 0.3.3 2020-01-21 [1] RSPM (R 4.2.0)
tibble * 3.2.1 2023-03-20 [1] RSPM (R 4.2.0)
tidybayes * 3.0.4 2023-03-14 [1] RSPM (R 4.2.0)
tidyr * 1.3.0 2023-01-24 [1] RSPM (R 4.2.0)
tidyselect 1.2.0 2022-10-10 [1] RSPM (R 4.2.0)
tidyverse * 2.0.0 2023-02-22 [1] RSPM (R 4.2.0)
timechange 0.2.0 2023-01-11 [1] RSPM (R 4.2.0)
tzdb 0.3.0 2022-03-28 [1] RSPM (R 4.2.0)
utf8 1.2.3 2023-01-31 [1] RSPM (R 4.2.0)
vctrs 0.6.2 2023-04-19 [1] RSPM (R 4.2.0)
withr 2.5.0 2022-03-03 [1] RSPM (R 4.2.0)
xfun 0.38 2023-03-24 [1] RSPM (R 4.2.0)
xtable 1.8-4 2019-04-21 [1] RSPM (R 4.2.0)
xts 0.13.1 2023-04-16 [1] RSPM (R 4.2.0)
yaml 2.3.7 2023-01-23 [1] RSPM (R 4.2.0)
zoo 1.8-12 2023-04-13 [1] RSPM (R 4.2.0)
[1] /usr/local/lib/R/site-library
[2] /usr/local/lib/R/library
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options(width = 80L)